Examples Illustrating the Use of Renormalization Techniques for Singularly Perturbed Differential Equations

Abstract

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Variation of Parameters and the Renormalization Group Method

“…The requirement of smoothness depends on the number of terms used in the asymptotic expansion. The renormalization group method has largely been developed by directly applying it as a loosely defined asymptotic technique to a variety of challenging examples found in the literature (cf., especially, Chen et al [1] and O'Malley and Kirkinis [2,3]). Alternative worthwhile discussions of the RG method include Ziane [4], DeVille et al [5], and Chiba [6].…”

“…We'll find it simpler to first convert the initial value problem for (2) to the form (1) for which we shall provide a simple renormalization method (cf. O'Malley and Williams [9]).…”

“…is the known unique solution of the limiting unperturbed initial value probleṁ φ = f (t, φ), t ≥ 0, φ(0) = A for a constant A. It then seeks a solution of the perturbed problem (2) in the form…”

A Combined Renormalization Group‐Multiple Scale Method for Singularly Perturbed Problems

“…e.g. [9]; [10,11], Chiba [12], and O'Malley and Kirkinis [13]) provides a successful alternative approach to solve many related problems. It proceeds by finding a regular perturbation solution or naive expansion, and then eliminating its secular terms by replacing integration constants with slowly varying functions, thereby resulting in a renormalized (i.e., uniformly valid) expansion.…”

Singular perturbed renormalization group theory and its application to highly oscillatory problems

“…So far, it has been proven to be very practicable in a large number of singular perturbed problems, such as secular problem, boundary layer problem, center manifold problem etc. [5,7,8,9,12,15,16,19].…”